Lecture 6 – Orthonormal Wavelet Bases David Walnut Department of Mathematical Sciences George Mason University Fairfax, VA USA Chapman Lectures, Chapman University, Orange, CA 6-10 November 2017 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Outline Wavelet systems Example 1: The Haar Wavelet basis Characterizations of wavelet bases Example 2: The Shannon Wavelet basis Example 3: The Meyer Wavelet basis Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Wavelet systems Definition A wavelet system in L2(R) is a collection of functions of the form j/2 j D2j Tk j,k Z = 2 (2 x k) j,k Z = j,k j,k Z { } 2 { − } 2 { } 2 where L2(R) is a fixed function sometimes called the 2 mother wavelet. A wavelet system that forms an orthonormal basis for L2(R) is called a wavelet orthonormal basis for L2(R). Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases If the mother wavelet (x) is concentrated around 0 then j j,k (x) is concentrated around 2− k. If (x) is essentially supported on an interval of length L, then j,k (x) is essentially j supported on an interval of length 2− L. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Since (D j T ) (γ)=D j M (γ) it follows that if is 2 k ^ 2− k concentrated on the interval I then j,k is concentrated on the interval 2j I. b b d Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Dyadic time-frequency tiling Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Example 1: The Haar System This is historically the first orthonormal wavelet basis, described by A. Haar (1910) as a basis for L2[0, 1]. The Haar basis is an alternative to the traditional Fourier basis but has the property that the partial sums of the series expansion of a continuous function f converges uniformly to f . We will prove “by hand” that the Haar basis is an orthonormal basis using a technique that will be generalized in the next lecture. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases The dyadic intervals Definition Define the dyadic intervals on R by = [2j k, 2j (k + 1)]: j, k Z , D { 2 } j j We write Ij,k =[2 k, 2 (k + 1)], and refer to the collection Ij,k : k Z { 2 } ` r ` as the dyadic intervals at scale j. Ij,k = Ij,k Ij,k , where Ij,k and r [ Ij,k are dyadic intervals at scale j + 1, to denote the left half and ` right half of the interval Ij,k . In fact, Ij,k = Ij+1,2k and r Ij,k = Ij+1,2k+1. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Note that given Ij,k , I = I I j,k j+1,2k [ j+1,2k+1 where Ij+1,2k is the left half of Ij,k and Ij+1,2k+1 the right half of Ij,k . Given (j , k ) =(j , k ), either I and I are disjoint or 0 0 6 1 1 j1,k1 j0,k0 one is contained in the other. In the latter case, the smaller interval is contained in either the right half or left half of the larger. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases The Dyadic step functions Definition A dyadic step function (at scale j) is a step function in L2(R) constant on the dyadic intervals Ij,k , k Z. We denote the 2 collection of all such step functions by Vj . For each j Z, Vj is a linear space. 2 If f V then f V for all j j, that is, V V if j > j. 2 j 2 j0 0 ≥ j ✓ j0 0 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases The Haar scaling function Definition Let p(x)=1[0,1)(x), and for each j, k Z, let 2 j/2 j p (x)=2 p(2 x k)=D j T p(x). j,k − 2 k The collection pj,k (x): j, k Z { 2 } is the system of Haar scaling functions. For fixed j, pj,k (x) k Z { } 2 is the system of scale j Haar scaling functions. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases j/2 Note that pj,k = 2 1I and that for each j, pj,k : k Z is j,k { 2 } an orthonormal system. j/2 In fact pj,k = 2 1Ij,k is an orthonormal basis for Vj . Since V V , we can write p in terms of the basis j ✓ j+1 j,k pj+1,k : k Z , and in fact { 2 } 1/2 1/2 pj,k = 2 pj+1,2k + 2 pj+1,2k+1. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases The Haar system Definition Let h(x)=1[0,1/2)(x) 1[1/2,1)(x), and for each j, k Z, let − 2 j/2 j h (x)=2 h(2 x k)=D j T h(x). j,k − 2 k The collection hj,k (x) j,k Z { } 2 is the Haar system on R. For fixed j, hj,k (x) k Z { } 2 is the system of scale j Haar functions. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Note that supp h = I and h V . j,k j,k j,k 2 j+1 Also h = p p . j,k j+1,2k − j+1,2k+1 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Theorem The Haar system is an orthonormal system in L2(R). If k = k 0 Z then hj,k and hj,k have disjoint supports. 6 2 0 If j < j0 then either hj,k and hj0,k 0 have disjoint supports, or supp hj0,k 0 is contained in an interval on which hj,k is constant. In either case, h , h = 0. h j0,k 0 j,k i Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Completeness of the Haar system Lemma For j Z, define the space Wj by 2 Wj = span hj,k : k Z . { 2 } Then V = V W . j+1 j ⊕ j Proof: ak pj+1,k = (a2n pj+1,2n + a2n+1 pj+1,2n+1) k Z n Z X2 X2 a + a = 2n 2n+1 (p + p ) 2 j+1,2n j+1,2n+1 n Z X2 a a + 2n − 2n+1 (p p ) 2 j+1,2n − j+1,2n+1 a2n + a2n+1 a2n a2n+1 = p , + − h , . 21/2 j n 21/2 j n n Z n Z X2 X2 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Completeness of the Haar system Theorem The Haar system is an orthonormal basis for L2(R). Proof: Define the orthogonal projection operators P f = f , p p . j h j,k i j,k k Z X2 Because any L2 function can be approximated by a dyadic step function, Pj f f as j , and also P j f 0 as ! !1 − ! j . !1 J 1 − Note that VJ = Wj V J which implies that ⊕ − j= J+1 M− 2 L (R)= Wj . j Z M2 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Smooth wavelet bases The Haar basis is the paradigmatic wavelet basis with good time localization. Since h(x) is discontinuous, it has very poor frequency localization. One of Meyer’s early contributions to the theory was to create an orthonormal wavelet basis with good decay in both time and frequency. Before getting to that proof, we will describe the paradigmatic orthonormal wavelet basis with optimal frequency localization, known as the Shannon basis. This will again be a proof “by hand” and will give the flavor of Meyer’s original proof. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Characterization of orthonormal bases Lemma A sequence x in a Hilbert space H is an orthonormal basis { n} for H if and only if 1. for all x H, x, x 2 = x 2, and 2 n |h ni| k k 2. x = 1 for all n. k nk P In other words, x is a normalized tight frame for H with { n} frame bound 1. The proof is trivial: Given m, note that 1 = x 2 = x , x 2 = x , x 2 + 1. k mk |h m ni| |h m ni| n n=m X X6 Hence x , x = 0 if n = m and x is an orthonormal h m ni 6 { n} system. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Characterization of wavelet ONB Theorem (A.) Suppose that L2(R) satisfies = 1. Then 2 k k2 j/2 j 2 (2 x k): j, k Z = j,k : j, k Z { − 2 } { 2 } is an orthonormal wavelet basis for L2(R) if and only if 1. (2j γ) 2 1, and | | ⌘ j Z X2 1 b 2. (2j γ) (2j (γ + k)) = 0 for all odd integers k. Xj=0 b b Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Characterization of wavelet ON systems Theorem (B.) 2 Given L (R), the wavelet system j,k j,k Z is an 2 { } 2 orthonormal system in L2(R) if and only if 1. (γ + k) 2 1, and | | ⌘ k Z X2 b 2. (2j (γ + k)) (γ + k)=0 for all j 1. ≥ k Z X2 b b Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Corollary 2 2 Given L (R), if j,k j,k Z is an orthonormal basis for L (R) 2 { } 2 then 1. (2j γ) 2 1, and | | ⌘ j Z X2 2. b(γ + k) 2 1. | | ⌘ k Z X2 b This says that a wavelet orthonormal basis must form a partition of unity in frequency both by translation and dilation.